Finite Group Databasebeta

← #30598377

Subgroups — #30598377

Order 15616.

780 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461722C212yesmaximal, complemented
2#21461680C211yesmaximal, characteristic
3#21461258C212yesmaximal, complemented
4#21461056C211yesmaximal, characteristic
5C2×#16840450C211yesmaximal, characteristic
6C2×#14519836—21no—
7D61⋊(C22⋊C8)C2212yes—
8#16840704C2212yes—
9#16840640—21no—
10#16840635—24no—
11#16840615—24no—
12#16840607—21no—
13#16840606—21no—
14#16840450—24no—
15#16840450—24no—
16C2×#14519772—21no—
17C2×#14519768—21no—
18C2×#14519762—21no—
19C2×#14519757C411yescharacteristic, complemented
20C2×#14519757C2211yescharacteristic
21C2×C4×Dic122C411yescharacteristic, complemented
22C2×C4×D244C2211yescharacteristic
23C2×#14519735—21no—
24C2×#14519735C411yescharacteristic, complemented
25C2×C4⋊C4×D61C2211yescharacteristic
26C2×#125×C61C411yescharacteristic, complemented
27C2×D61⋊C8—22no—
28C2×C4⋊#6923—22no—
29C2×C22⋊#6923—22no—
30#14519836—42no—
31#14519834—22no—
32C22⋊#92420—22no—
33(C2×#6923)⋊C4—22no—
34C22×Dic122C2×C411yescharacteristic
35C22×D244D411yescharacteristic
36C22×C4×D61—21no—
37C22×C4×D61C2×C411yescharacteristic
38C22×C4×D61C2311yescharacteristic
39#14519772—28no—
40#14519768—28no—
41#14519762—28no—
42#14519757—22no—
43#14519757—22no—
44#14519757—22no—
45#14519757D414yes—
46C4×Dic122—22no—
47C4×Dic122—22no—
48C4×D244—22no—
49C4×D244—22no—
50#14519735—28no—
51#14519735—22no—
52#14519735—22no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×C4⋊Dic61D411yescharacteristic
57C2×#11758196—21no—
58C2×#11758196—21no—
59C4⋊C4×D61—22no—
60C4⋊C4×D61D414yes—
61C2×#11758198—21no—
62C2×#11758198—21no—
63C2×#11758198—21no—
64C2×#11758198D411yescharacteristic
65C2×#11758198D411yescharacteristic
66C2×C4×Dic61—21no—
67C2×C4×Dic61—21no—
68#125×C61—22no—
69#125×C61—22no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61C2×C411yescharacteristic
73C2×C4⋊C4×C61C2×C411yescharacteristic
74C2×C42×C61C2×C411yescharacteristic
75C23×D61C22⋊C411yescharacteristic
76C23×D61C2×D411yescharacteristic
77C22×#6923—42no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22×C411yescharacteristic
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61C22⋊C411yescharacteristic
86C22×Dic61C2×D411yescharacteristic
87C2×D244—22no—
88C2×D244—22no—
89C2×D244—21no—
90C2×D244—21no—
91C2×C4×D61—24no—
92C2×C4×D61—21no—
93C2×C4×D61—21no—
94C2×C4×D61—42no—
95C2×C4×D61—42no—
96C2×C4×D61—41no—
97C2×C4×D61—41no—
98C2×C4×D61—42no—
99C2×C4×D61C2×D412yes—
100C2×C4×D61C2×D412yes—
101C2×Dic122—21no—
102C2×Dic122—42no—
103C2×Dic122C22⋊C412yes—
104C22⋊#6923—42no—
105C22⋊#6923—42no—
106C2×#92420—42no—
107C4⋊#6923—44no—
108D61⋊C8—44no—
109C4⋊C4×C61—24no—
110C4⋊C4×C61—24no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61C22⋊C412yes—
114C4⋊C4×C61C22⋊C412yes—
115C42×C61—21no—
116C42×C61—21no—
117#11758198—24no—
118#11758198—24no—
119#11758198—24no—
120#11758198—22no—
121#11758198—22no—
122C4⋊Dic61—24no—
123C4⋊Dic61—21no—
124C4⋊Dic61—24no—
125C4⋊Dic61—22no—
126C4⋊Dic61C22⋊C412yes—
127#11758196—24no—
128#11758196—24no—
129C4×Dic61—24no—
130C4×Dic61—24no—
131C23×C61C2×C22⋊C411yescharacteristic
132C22×D61—21no—
133C22×D61—21no—
134C22×D61—21no—
135C22×D61—41no—
136C22×D61—41no—
137C22×D61—41no—
138C22×D61—41no—
139C22×D61#12012yes—
140C22×D61#13512yes—
141C22×D61AGL(1,ℤ/8)11yescharacteristic
142C22×D61#13711yescharacteristic
143C2×#6923—82no—
144C2×#6923—82no—
145C2×#6923—82no—
146C2×C4×C61—22no—
147C2×C4×C61—21no—
148C2×C4×C61—22no—
149C2×C4×C61—21no—
150C2×C4×C61—22no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—42no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61C2×C22⋊C411yescharacteristic
158C2×C4×C61C2×C22⋊C411yescharacteristic
159C2×Dic61—22no—
160C2×Dic61—21no—
161C2×Dic61—22no—
162C2×Dic61—41no—
163C2×Dic61—42no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61#12012yes—
168D244—42no—
169D244—42no—
170C4×D61—24no—
171C4×D61—42no—
172C4×D61—42no—
173C4×D61—44no—
174C4×D61—44no—
175C4×D61—42no—
176Dic122—42no—
177Dic122—42no—
178#92420—82no—
179#8382—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61#37911yescharacteristic
183C22×C61#41611yescharacteristic
184C22×C61#41711yescharacteristic
185C2×D61—22no—
186C2×D61—22no—
187C2×D61—42no—
188C2×D61—41no—
189C2×D61—41no—
190C2×D61—41no—
191C2×D61—41no—
192C2×D61—41no—
193C2×D61—81no—
194C2×D61—81no—
195C2×D61#44912yes—
196C2×D61#45212yes—
197#6923—162no—
198C4×C61—21no—
199C4×C61—41no—
200C4×C61—42no—
201C4×C61—42no—
202C4×C61—41no—
203C4×C61#41012yes—
204Dic61—22no—
205Dic61—42no—
206Dic61—41no—
207Dic61—42no—
208C2×#552—611no—
209#1823—612no—
210#1799—611no—
211#1520—612no—
212#1459—611no—
213C2×C61—21no—
214C2×C61—21no—
215C2×C61#178211yescharacteristic
216C2×C61#178811yescharacteristic
217C2×C61#179611yescharacteristic
218D61—22no—
219D61—22no—
220D61—81no—
221#552—1224no—
222#552—1224no—
223C2×#126—1221no—
224C2×#125—1221no—
225C2×#125#692311yescharacteristic, complemented
226C2×#123—1221no—
227C2×C22⋊Q8—611no—
228C2×C22⋊Q8—611no—
229C2×C4×Q8—611no—
230C2×C4×D4—611no—
231C2×#117—611no—
232C2×#117—1221no—
233C22×C4⋊C4—611no—
234C2×#99—1221no—
235C2×C22⋊C8—612no—
236#380—612no—
237#342—1221no—
238#340—1224no—
239#328—1224no—
240#324—1221no—
241#323—1221no—
242C61#838211yescharacteristic, complemented, Sylow
243C22×Q8—611no—
244C22×D4—611no—
245C23×C4—611no—
246C23×C4—611no—
247C23×C4—1221no—
248C22×C8—1222no—
249#126—1228no—
250#125—22no—
251#125—22no—
252#125—1228no—
253#123—1228no—
254C22⋊Q8—614no—
255C22⋊Q8—1222no—
256C22⋊Q8—1222no—
257C22⋊Q8—1222no—
258C4×Q8—1222no—
259C4×Q8—1222no—
260C4×D4—1222no—
261C4×D4—1222no—
262#117—1222no—
263#117—1228no—
264#117—1222no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—611no—
268C2×C4⋊C4—614no—
269C2×C4⋊C4—611no—
270C2×C4⋊C4—1222no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1222no—
276C2×C4⋊C4C2×#692311yescharacteristic
277C2×C4⋊C4C2×#692311yescharacteristic
278C2×C22⋊C4—611no—
279C2×C22⋊C4—611no—
280C2×C22⋊C4—1221no—
281C2×C22⋊C4—1221no—
282C2×C22⋊C4—1221no—
283C2×C22⋊C4—1222no—
284C2×C42—1221no—
285C2×C42—1221no—
286C2×C42C2×#692311yescharacteristic
287#99—2442no—
288C22⋊C8—1222no—
289C22⋊C8—1222no—
290#95—1222no—
291C24—611no—
292C24—611no—
293C2×Q8—612no—
294C2×Q8—1221no—
295C2×Q8—2442no—
296C2×D4—1221no—
297C2×D4—1222no—
298C2×D4—1222no—
299C2×D4—1221no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—611no—
303C22×C4—612no—
304C22×C4—612no—
305C22×C4—611no—
306C22×C4—1224no—
307C22×C4—1221no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—1221no—
311C22×C4—2442no—
312C22×C4—2442no—
313C22×C4—2442no—
314C22×C4—2441no—
315C22×C4—2441no—
316C22×C4—2442no—
317C22×C4C22⋊#692311yescharacteristic
318C22×C4C22⋊#692311yescharacteristic
319C22×C4C22×#692311yescharacteristic
320C2×C8—2442no—
321C2×C8—2444no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—612no—
327C4⋊C4—1224no—
328C4⋊C4—1222no—
329C4⋊C4—1224no—
330C4⋊C4—1224no—
331C4⋊C4—1224no—
332C4⋊C4—1221no—
333C4⋊C4—2444no—
334C4⋊C4C22⋊#692312yes—
335C4⋊C4C22⋊#692312yes—
336C22⋊C4—1222no—
337C22⋊C4—1222no—
338C22⋊C4—1224no—
339C22⋊C4—1224no—
340C22⋊C4—1224no—
341C22⋊C4—2442no—
342C22⋊C4—2442no—
343C42—21no—
344C42—21no—
345C42—1224no—
346C42—1224no—
347C23—612no—
348C23—612no—
349C23—611no—
350C23—611no—
351C23—1221no—
352C23—1221no—
353C23—1221no—
354C23—2441no—
355C23—2441no—
356C23—2441no—
357C23—2441no—
358C23C2×C22⋊#692311yescharacteristic
359Q8—2442no—
360Q8—2442no—
361D4—2442no—
362D4—2442no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—612no—
375C2×C4—1224no—
376C2×C4—1222no—
377C2×C4—1221no—
378C2×C4—1222no—
379C2×C4—4882no—
380C2×C4—4882no—
381C2×C4—4882no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4—2444no—
385C2×C4—2444no—
386C2×C4—2441no—
387C2×C4—2442no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2442no—
391C2×C4—2441no—
392C2×C4C2×C22⋊#692311yescharacteristic
393C2×C4C2×C22⋊#692311yescharacteristic
394C8—4882no—
395C22—21no—
396C22—21no—
397C22—612no—
398C22—612no—
399C22—1222no—
400C22—1222no—
401C22—4881no—
402C22—4881no—
403C22—2442no—
404C22—2441no—
405C22—2441no—
406C22—2441no—
407C22—2441no—
408C22—2441no—
409C22#1684070311yescharacteristic
410C22#1684075211yescharacteristic
411C22#1684075311yescharacteristic
412C4—21no—
413C4—41no—
414C4—42no—
415C4—41no—
416C4—42no—
417C4—1222no—
418C4—9762no—
419C4—2441no—
420C4—2442no—
421C4—2442no—
422C4#1684074312yes—
423C2—21no—
424C2—21no—
425C2—1222no—
426C2—1222no—
427C2—4881no—
428C2#2146165011yescharacteristic
429C2#2146165911yescharacteristic
430C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#125×C61C4
socleC22×C61#379
prime radicalC22×D244D4
cosocleC22×C4×D61C23
hypercenterC2×#125#6923
hypocenterC61#8382
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#125

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21461722C2semidirect
3. #21461258C2semidirect
19. C2×#14519757C4semidirect
21. C2×C4×Dic122C4semidirect
24. C2×#14519735C4semidirect
26. C2×#125×C61C4semidirect
225. C2×#125#6923semidirect
242. C61#8382semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214617222
#214616802
#214612582
#214610562
C2×#168404502
#838261